Search results
Results from the WOW.Com Content Network
The long real line pastes together ℵ 1 * + ℵ 1 copies of the real line plus a single point (here ℵ 1 * denotes the reversed ordering of ℵ 1) to create an ordered set that is "locally" identical to the real numbers, but somehow longer; for instance, there is an order-preserving embedding of ℵ 1 in the long real line but not in the real ...
An axiomatic definition of the real numbers consists of defining them as the elements of a complete ordered field. [2] [3] [4] This means the following: The real numbers form a set, commonly denoted , containing two distinguished elements denoted 0 and 1, and on which are defined two binary operations and one binary relation; the operations are called addition and multiplication of real ...
In summary, a set of the real numbers is an interval, if and only if it is an open interval, a closed interval, or a half-open interval. [4] [5] A degenerate interval is any set consisting of a single real number (i.e., an interval of the form [a, a]). [6] Some authors include the empty set in this definition.
This article lists some properties of sets of real numbers. The general study of these concepts forms descriptive set theory , which has a rather different emphasis from general topology . Definability properties
The set of real numbers has several standard structures: An order: each number is either less than or greater than any other number. Algebraic structure: there are operations of addition and multiplication, the first of which makes it into a group and the pair of which together make it into a field .
For the case n = 2, an extension of the Euclidean algorithm can find any integer relation that exists between any two real numbers x 1 and x 2.The algorithm generates successive terms of the continued fraction expansion of x 1 /x 2; if there is an integer relation between the numbers, then their ratio is rational and the algorithm eventually terminates.
Unlike rational number arithmetic, real number arithmetic is closed under exponentiation as long as it uses a positive number as its base. The same is true for the logarithm of positive real numbers as long as the logarithm base is positive and not 1. [105] Irrational numbers involve an infinite non-repeating series of decimal digits.
The algebra of sets is the set-theoretic analogue of the algebra of numbers. Just as arithmetic addition and multiplication are associative and commutative, so are set union and intersection; just as the arithmetic relation "less than or equal" is reflexive, antisymmetric and transitive, so is the set relation of "subset".